Four Turyn type sequences of lengths 36, 36, 36, 35 are found by a computer search. These sequences give new base sequences of lengths 71, 71, 36, 36 and are used to generate a number of new T-sequences. The first order of many new Hadamard matrices constructible using these new T-sequences is 428. #

A divisible design graph is a graph whose adjacency matrix is the incidence matrix of a divisible design. Divisible design graphs are a natural generalization of (v, k, λ)-graphs, and like (v, k, λ)-graphs they make a link between combinatorial design theory and algebraic graph theory. The study of divisible design graphs benefits from, and contributes to, both parts. Using information of the eigenvalues of the adjacency matrix, we obtain necessary conditions for existence. Old results of Bose and Connor on symmetric divisible designs give other conditions and information on the structure. Many constructions are given using various combinatorial structures, such as (v, k, λ)-graphs, distance-regular graphs, symmetric divisible designs, Hadamard matrices, and symmetric balanced generalized weighing matrices. Several divisible design graphs are characterized in terms of the parameters.

The class of mutually unbiased Hadamard (MUH) matrices is studied. We show that the number of MUH matrices of order 4n 2 , n odd is at most 2 and that the bound is attained for n = 1, 3. Furthermore, we find a lower bound for the number of MUH matrices of order 16n 2 , assuming the existence of a Hadamard matrix of order 4n. An extension to unbiased weighing matrices is also presented.

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